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Theorem symdif1 3230
Description: Two ways to express symmetric difference. This theorem shows the equivalence of the definition of symmetric difference in [Stoll] p. 13 and the restated definition in Example 4.1 of [Stoll] p. 262. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
symdif1 ((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐴𝐵) ∖ (𝐴𝐵))

Proof of Theorem symdif1
StepHypRef Expression
1 difundir 3218 . 2 ((𝐴𝐵) ∖ (𝐴𝐵)) = ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))
2 difin 3202 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
3 incom 3157 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
43difeq2i 3087 . . . 4 (𝐵 ∖ (𝐴𝐵)) = (𝐵 ∖ (𝐵𝐴))
5 difin 3202 . . . 4 (𝐵 ∖ (𝐵𝐴)) = (𝐵𝐴)
64, 5eqtri 2076 . . 3 (𝐵 ∖ (𝐴𝐵)) = (𝐵𝐴)
72, 6uneq12i 3123 . 2 ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) = ((𝐴𝐵) ∪ (𝐵𝐴))
81, 7eqtr2i 2077 1 ((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐴𝐵) ∖ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:   = wceq 1259  cdif 2942  cun 2943  cin 2944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952
This theorem is referenced by: (None)
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